To research the shorter circuit cover in a bridgeless graph, Fan conjectured that if G/C admits a nowhere-zero 4-flow, then G admits a 4-flow f such that \(E(G)-E(C)\subseteq \text {supp} (f)\) and \(|\text {supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|\) , where G is a bridgeless graph and C is a circuit of G, and himself confirmed the conjecture for \(|E(C)|\le 19\) (Fan in Combinatorica 37(6):1097–1112, 2017). Wang et al. showed that the conjecture holds for \(|E(C)|\le 27\) (Wang et al. in Acta Math Sin Engl Ser 38(9):1653–1664, 2022). In this paper, we prove that the conjecture holds for \(|E(C)|\le 35\) .
